In April's feature story on location science, professor
Charles ReVelle invited readers to match wits with Emperor
Constantine in tackling one of the world's oldest recorded
location problems ("Can
You Protect the Roman Empire?" p.40). Below, Professor
ReVelle lays out possible solutions:

To review, a region was secured if it already had a pebble (six
legions) within its borders. Alternatively, a region was
securable if a pebble could reach the region in just one step,
and that could occur only if somewhere, within one step of the
region, there were one or more places that had two pebbles
garrisoned. That is, a pebble could only be "launched" to its
destination by another pebble.

The problem we pictured last time was meant to get you "warmed
up" for greater challenges. There is, looking exhaustively at the
possibilities for Figure 1, just one alternative that secures or
makes securable all the regions of the empire. That single
alternative, as shown in Figure 1, places two pebbles at Rome, as
well as one pebble in Britain and one in Asia Minor. With this
deployment, Britain, Asia Minor, and Rome are secured and all
other regions can be reached by a pebble launched from Rome. It
is a "Roman Solution"; that is, Rome, the heart of the empire, is
not without forces at its disposal.

That alternative, of course, is different from the choice that
Constantine made. Constantine placed two pebbles at Rome and two
pebbles at Constantinople, securing or making securable all but
Britain--which he, not surprisingly, ended up losing. Why, then,
did Constantine pursue this strategy?

Probably Constantine excluded the apparently stronger alternative
because it was not "robust." By this we mean that, although the
deployment protected all regions in the event of a first war, it
lacked the capacity to perform well if, elsewhere in the empire,
the forces of insurrection or invasion opened a second front. To
see why this is so, consider the solution (2-R, 1-B, 1-AM) in
place, and imagine a first war breaking out anywhere in the
empire that a pebble does not currently reside. This could be in
Gaul, Iberia, North Africa, Egypt, or Constantinople. Suppose,
for example, the war is in Gaul and a pebble is dispatched from
Rome to Gaul, where it now must reside for a time while order is
restored. With this pebble deployed, how many regions will now
be unprotected? The other four regions previously protected by
the two-pebble unit in Rome.

This suggests the need to place two-pebble units in such a way as
to double-cover regions--so that they are still protected in the
event of a second war. In sum, the described alternative (2-R,
1-B, 1-AM), though it appeared strong at first glance, can now be
seen to have a serious weakness: an inability to respond to a
second war. To correct for this weakness, we need to expand our
thinking to alternatives that take into consideration readiness
in the event of a second war.

Before we go on, however, we have to explain why we called this
problem a "warm-up." We had to get you thinking about
alternatives and show how a deployment that looked good initially
really wasn't so strong. Now we come to the tougher problem.

In Figure 2, the Roman Empire looks slightly different than in
Figure 1. The same regions are shown, but two further
connections are displayed--one between Iberia and Britain, the
other between Egypt and Asia Minor. The problem shown in Figure
2 is, in fact, the real Roman Empire. These two further
connections complicate the problem since they introduce new
possibilities for coverage. The deployment (2-R, 1-B, 1-AM) still
protects the entire empire in the event of a first war, so it is
still a solution. But now, the new connections make possible
other deployments that also protect the entire empire.

What are these other deployments that protect the entire empire?
Which of these does the best in the event of a second war? Does
Constantine's solution look any better than before?

We'll feature a thorough discussion of the expanded Roman
Legions problem in September's web version of the
magazine.